In ([1]}), the authors dealt with the Question REF . Let \(P(z)\)  be a non-constant monic polynomial. We say \(P(z)\)  as a uniqueness polynomial if \(P(f)\equiv P (g)\)  implies \(f \equiv g\)  for any two non-constant meromorphic functions \(f,~g\) ; while a strong uniqueness polynomial if \(P(f)\equiv cP (g)\)  implies \(f \equiv g\)  for any two non-constant meromorphic functions \(f,~g\)  and non-zero constant \(c\) . Moreover, we say that \(P(z)\)  is satisfying the Fujimoto's Hypothesis ([2]})) if
\(P(\beta _{l_{s}})\ne P(\beta _{l_{t}})~~~~(1\le l_{s}< l_{t}\le k).\) 
